Determine how many solutions exist for the system of equations. ${-x+y = -7}$ ${x-y = 9}$
Solution: Convert both equations to slope-intercept form: ${-x+y = -7}$ $-x{+x} + y = -7{+x}$ $y = -7+x$ ${y = x-7}$ ${x-y = 9}$ $x{-x} - y = 9{-x}$ $-y = 9-x$ $y = -9+x$ ${y = x-9}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = x-7}$ ${y = x-9}$ Both equations have the same slope with different y-intercepts. This means the equations are parallel. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Parallel lines never intersect, thus there are NO SOLUTIONS.